Who Keeps The Money You Found On The Floor?

Welcome to The Riddler. Every week, I offer up a problem related to the things we hold dear around here: math, logic and probability. These problems, puzzles and riddles come from many top-notch puzzle folks around the world — including you! You’ll find this week’s puzzle below.

Mull it over on your commute, dissect it on your lunch break and argue about it with your friends and lovers. When you’re ready, submit your answer using the link below. I’ll reveal the solution next week, and a correct submission (chosen at random) will earn a shoutout in this column. Important small print: To be eligible, I need to receive your correct answer before 11:59 p.m. EDT on Sunday. Have a great weekend!

Before we get to the new puzzle, let’s return to last week’s. Congratulations to 👏 Lisa Duncan 👏 of Redwood City, California, our big winner. You can find a solution to the previous Riddler at the bottom of this post.

Now here’s this week’s Riddler, a parlor game puzzle that comes to us from Bruce Torrence, a math professor at Randolph Macon College in Ashland, Virginia.

You and four statistician colleagues find a $100 bill on the floor of your department’s faculty lounge. None of you have change, so you agree to play a game of chance to divide the money probabilistically. The five of you sit around a table. The game is played in turns. Each turn, one of three things can happen, each with an equal probability: The bill can move one position to the left, one position to the right, or the game ends and the person with the bill in front of him or her wins the game. You have tenure and seniority, so the bill starts in front of you. What are the chances you win the money?

Extra credit: What if there were N statisticians in the department?

Submit your answer
Need a hint? You can try asking me nicely. Want to submit a new puzzle or problem? Email me. I’m especially on the hunt for Riddler Express problems — bite-size puzzles that don’t take quite as much time or computational power to solve.

And here’s the solution to last week’s Riddler, concerning your leadership of a banana-hungry camel on his way to a far-flung bazaar. If you start with 3,000 bananas, are guiding a camel that can carry only 1,000 bananas at a time and eats one of them every mile it walks, and are going to a bazaar 1,000 miles away, the best you can do is sell 833 bananas at the market.

Clearly, you don’t do well simply loading up the camel to capacity and heading straight to market. By the time you got there, the camel would have eaten all the bananas. (He’d eat a banana every mile for the 1,000-mile journey.) You need a subtler strategy. Essentially, you should create banana way stations, where you deposit caches of fruit along your route. It’s most efficient to fully load up your camel at the beginning of each trip, so you should aim to create two way stations at which you’ll eventually drop off 2,000 and 1,000 bananas.

Here’s how the math works out: Take your 3,000 bananas, and move them all to a spot one-third of the way to the bazaar, in three trips. Given the camel’s voracious per-mile appetite, this leaves you with 2,000 uneaten bananas at that way station. Take those 2,000 bananas another 500 miles in two trips. This leaves you with 1,000 bananas and 166 ⅔ miles to go. Take one more trip with those bananas to the bazaar. You’ll arrive with 833 ⅓ uneaten bananas. (And you’ll probably only be able to sell 833 of them, because who is going to buy a banana mostly eaten by a camel?)

Laurent Lessardplotted the general case where N bananas need to be transported D miles:

But the 🏆 Coolest Riddler Extension Award 🏆 this week goes to Diarmuid Early. He summed up his capitalistic extension in the following image:

If there are many other camel owners, each with his own camel and bananas, you’d do well to hire some of them. Specifically, Diarmuid finds, you need to hire nine employees before you can stop traveling yourself.